1 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics

Transcription

1 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 3 1 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics Jack F. Fowler CONTENTS Glossary Introduction The Simplest Modeling The Seven Steps to LQ Heaven Brief Summary The Seven Steps to LQ Heaven The Details Development of the Simple LQ Formula E = nd (1 + d / / ) Biologically Effective Dose Relative Effectiveness Overall Treatment Time Acute Mucosal Tolerance To Convert from BED to NTD or EQD 2 Gy One Example What is the Standard of Precision of These Estimates of BED or NTD? Gamma Slopes Rejoining Point for Those Who Skipped: How to Evaluate a New Schedule Brief Summary Now Let Us Study Some of the Best-Known Schedules for Head-And-Neck Tumor Radiotherapy Standard Fractionation Hyperfractionation Radiation Therapy Oncology Group Four-Arm Fractionation Trial (RTOG 90-03) Head-and-Neck Schedules That Were Initially Too Hot in Table Shortening the Wang 2-Fraction-a-Day Schedule Using BED to Adjust Individual Doses General Considerations of Head-and-Neck Radiotherapy A Theoretical Calculation of Close to Optimum Head & Neck Schedules: 3 Weeks at Five Fractions per Week Conclusions Re Head-and-Neck Schedules Concurrent Chemotherapy Hypofractionation for Prostate Tumors Summary Appendix: Is This a Mistaken Dose Prescription? For Equal Late Complications For Equal Tumor Effect Acute Mucosal Effects Line-by-Line Worked Examples: Details of Calculations of the Schedules Discussed in This Appendix 27 References 29 This chapter is written mainly for those who say I don t understand this business I can t be bothered with Linear Quadratic and that sort of stuff. Well, it might seem boring depending on your personality but it is easy, and it makes so many things in radiation therapy wonderfully and delightfully clear. Experienced readers can turn straight to Section 1.4. J. F. Fowler, DSc, PhD Emeritus Professor of Human Oncology and Medical Physics, Medical School of University of Wisconsin, Madison, Wisconsin, USA; Former Director of the Gray Laboratory, Northwood, London, UK Present address: 150 Lambeth Road, London, SE1 7DF, UK Glossary, alpha Intrinsic radiosensitivity. Log e of the number of cells sterilized non-repairably per gray of dose of ionizing radiation., beta Repair capacity. Log e of the number of cells sterilized in a repairable way per gray squared., alpha / beta ratio the ratio of intrinsic radiosensitivity to repair capability of a specified tissue. This ratio is large (>8 Gy) for rapidly proliferating tissues and most tumors. It is small (<6 Gy) for slowly proliferating tissues, including late normal-tissue complications. This difference is vital for the success of radiotherapy. When beta ( ) is large, both mis-repair and good-repair are high. It is the mis-repair that causes the cell survival curve to bend downward.

2 4 J. F. Fowler Accelerated fractionated schedules with shorter overall times than the conventional 7 (or 6) fractionation weeks. BED Biologically effective dose, proportional to log cell kill and therefore more conceptually useful as a measure of biological damage than physical dose, the effects of which vary with fraction size and dose rate. Formally, the radiation dose equivalent to an infinite number of infinitely small fractions or a very low dose-rate. Corresponds to the intrinsic radiosensitivity ( ) of the target cells when all repairable radiation damage ( ) has been given time to be repaired. In linear quadratic modeling, BED = total dose relative effectiveness (RE), where RE = (1 + d ), with d = dose per fraction, = intrinsic radiosensitivity, and = repair capacity of target cells. bned Biochemically no evidence of disease. No progressive increase of prostate specific antigen (PSA) level in patients treated for prostate cancer. CI Confidence interval (usually ±95%). CTV Clinical tumor volume. The volume into which malignant cells are estimated to have spread at the time of treatment, larger than the gross tumor volume (GTV) by at least several millimeters, depending on site, stage, and location. See also GTV and planning treatment volume (PTV). t Time interval between fractions, recommended to be not less than 6 h. EBR External beam radiation. EGFR Epithelial growth factor receptor, one of the main intracellular biochemical pathways controlling rate of cell proliferation. EQD Biologically equivalent total dose, usually in 2-Gy dose fractions. The total dose of a schedule using, for example, 2 Gy per fraction that gives the same log cell kill as the schedule in question. If so, should be designated by the subscript EQD2 Gy. EUD Equivalent uniform dose. A construct from the DVH of a non-uniformly irradiated volume of tissue or tumor that estimates the surviving proportion of cells for each volume element (voxel), sums them, and calculates that dose which, if given as a uniform dose to the same volume, would give the same total cell survival as the given non-uniform dose. Local fraction size is taken into account by assuming an ratio for the tissue concerned. Gamma, -50, -37 Slope of a graph of probability, usually tumor control probability (TCP), versus total fractionated dose (NTD or EQD), as percentage absolute increase of probability per 1% increase in dose. The steepest part of the curve is at 50% for logistic-type curves and at 37% for Poisson-type curves. Tumor TCP is usually between a gamma-50 (or -37) of 1.0 and 2.5. The difference between -50 and -37 is rarely clinically significant. Gy, gray The international unit of radiation dose: one joule per kilogram of matter. Commonly used radiotherapy doses are approximately 2 Gy on each of 5 days a week. Gy10, Gy3, Gy1.5 Biologically effective dose (BED), with the subscript representing the value of that tissue s ratio = 10 Gy for early radiation effects, 3 Gy for late radiation effects and 1.5 Gy for prostate tumors. The subscript confirms that this is a BED, proportional to log cell kill, and not a real physical dose. GTV Gross tumor volume. The best estimate of tumor volume visualized by radiological, computed tomography (CT) scan, magnetic resonance, ultrasound imaging, or positron emission tomography. HDR High dose rate. When the dose fraction is delivered in less than five or ten minutes; that is, much shorter than any half-time of repair of radiation damage. Hyperfractionation More (and smaller) dose fractions than 1.8 Gy or 2 Gy. Hypofractionation Fewer (and larger) dose fractions than 1.8 Gy or 2 Gy. Isoeffect Equal effect.

3 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 5 LC Local control (of tumors). LDR Low dose rate. Officially (ICRU), less than 2 Gy/h; but this is deceptive because any dose rate greater than 0.5 Gy/h will give an increased biological effect compared with the traditional 0.42 Gy/h (1000 cgy per day). For example at 2 Gy/h, the biological effects will be similar to daily fractions of 3.3 Gy and 2.8 Gy on late complications and on tumors respectively. Linear effect Directly proportional to dose. Ln log e Natural logarithm, to base e. One log 10 is equal to log e. Log 10 Common logarithm, to base 10. Ten logs of cell kill are log e of cell kill. LQ Linear quadratic formula: log e cells killed = dose + dose-squared. Logistic curve A symmetrical sigmoid or S-shaped graph relating the statistically probable incidence of events, including complications, or tumors controlled, at a specified time after treatment, to total dose (NTD). This curve is steepest at the probability of 50%. LRC Loco-regional tumor control. LC would be local control. NTCP Normal tissue complication probability. NTD Normalized total dose of any schedule. The total dose of a schedule using 2 Gy per fraction that gives the same log cell kill as the schedule in question. The NTD will be very different for late effects (with = 3 Gy and no overall treatment time factor) than for tumor effect (with = 10 Gy and an appropriate time factor). Poisson curve A near-sigmoid graph of probability of occurrence of events, such as tumor control at X years, versus total dose or NTD. Based on random chance of successes among a population of tumors or patients, the probability of curve P=exp ( n), where an average of n cells survive per tumor after the schedule, but 0 cells must survive to achieve 100% cure. If an average of 1 cell survives per tumor, P = 37%. If 2 cells survive, P = 14%. If 0.1 cells survive on average, P = 90%. This curve is steepest at the probability of 37%. PTV Planning treatment volume larger than CTV to allow for set-up and treatmentplanning errors. PSA Prostate-specific antigen: can be measured in a blood specimen as a measure of activity of the prostate gland. Often taken as a measure of activity of prostate cancer. Quadratic Effect proportional to dose squared, for example from two particle tracks passing through a target. QED Quod Erat Demonstrandum Latin for That s what we wanted to show! RE Relative effectiveness. We multiply total dose by RE to obtain BED. RE = (1 + d [ ]) where d is the dose per fraction. RTOG Radiation Therapy Oncology Group, USA. SF Surviving fraction after irradiation, usually of cells. Tpot Potential doubling time of cells in a population; before allowing for the cell loss factor. Tpot is the reciprocal of cell birth rate. It can only be measured in a tissue before any treatment is given to disturb its turnover time. Tp Cell doubling time in a tissue during radiotherapy; probably somewhat faster than Tpot. Determined from gross tumor (or other tissue) results when overall time is altered. Tk Kick-off or onset time: the apparent starting time of rapid compensatory repopulation in tumor or tissue after the start of treatment, when it is assumed that there are just two rates of cell proliferation during radiotherapy: zero from start to Tk, then constant doubling each Tp days until end of treatment at T days. Accelerating repopulation is discussed in Section TCP Tumor control probability.

4 6 J. F. Fowler 1.1 Introduction It is well known that the simplest description of radiation dose, the total dose, is not adequate because its effect varies with size of dose per session (the dose fraction) and with dose rate. If we double the dose per fraction from 2 Gy to 4 Gy (keeping total dose constant), the effect is 20% greater for tumors but 100% greater for late complications. Further, if a given physical dose is spread evenly over 24 h instead of 2 min, its effect is reduced by 20% for most tumors, but to about half for late complications. We need a way of expressing radiation dose in some quantitative way that is more proportional to the observed biological effect. This is the object of calculating a biologically effective dose (BED), and an equivalent dose in 2 Gy fractions (EQD or NTD), so that a 20% increase or decrease of BED or NTD or EQD 2 Gy will lead to a reasonable approximation of a 20% increase or decrease of the expected biological effect. The interesting point is that the same change in physical dose is likely to alter the incidence of late complications to double or half of its effect on tumors. So how can we deal with that? The basic truth in radiotherapy is that any change in the schedule of dose delivery has a different effect on tumors from its effect on late complications, unless both dose per fraction and dose rate are kept constant. These differences provide some of the remarkable advantages of radiation therapy, and also some puzzles until they are explained. BED can take these differences into account, and preferably explain them. In the 25 years since the linear quadratic (LQ) formula has been used for the evaluation of radiotherapy schedules, it has proved remarkably reliable. It is now the main and generally accepted method of rationalizing the improved time dose-fractionation schedules that have been developed to replace, in some body sites, the standard 2 Gy given five times a week for 6 or 7 weeks schedules. It was first of all useful in identifying the important difference in the effect of dose-fraction size between rapidly proliferating tissues (most tumors) and slowly proliferating tissues (most late complications). This explained the blindly used, but not always wrong, predominance of multi-small-fraction schedules, such as 1.8 or 2 Gy five times a week for 6 or 7 weeks. As explained below, the theoretically ideal overall time would be close to the time at which rapid repopulation in the tumor kicks off, designated Tk days after starting treatment. In the early years of the development of the LQ formulation, there was no overall treatment-time factor (Douglas and Fowler 1976; Barendsen 1982; Withers et al. 1983). This was added later (Travis and Tucker 1987; van de Geijn 1989; Fowler 1989), based on LQ-aided analyses of animal and clinical data (Denekamp 1973; Turesson and Notter 1984a,b; Thames and Hendry 1987). Since then, the strong effect of repopulation of tumor cells during radiotherapy has been well substantiated so that a repopulation term has been added for tumors (Fowler 1978, 1989; Withers et al. 1988; Fowler and Lindstrom 1992; Hendry et al. 1996). More recently, a different set of parameters has been described to predict acute mucosal reactions in human patients (Fowler et al. 2003c). Although the accuracy and even the nature of the LQ factors has been queried a few times, for example whether the parameters are unique or distributed (King and Mayo 2000; Brenner and Hall 1999, 2000; King and Fowler 2002; Dasu et al. 2003; Moiseenko 2004), the LQ formulation has remained solidly useful and has aided in the design of clinical trials that have changed the practice of radiotherapy (Thames et al. 1983). Examples include the design of hyperfractionated (more and smaller fractions) and accelerated fractionation (shorter) trials, the avoidance of gaps in radiation treatment, the development of high dose-rate brachytherapy, and a better understanding of when to use or avoid hypofractionation (fewer and larger fractions). The recent growth of stereotactic body radiotherapy is a subset of the latter category (Fowler et al. 2004). One of the most interesting series of modeling investigations concerns oral and laryngeal cancers in which the overall times were deliberately shortened until the acute reactions became too severe, in several well known schedules in different countries. Each schedule was then moderated in some way until it became tolerable. The modeling then showed that not only the acute mucosal reactions fell into a narrow band of BED, but the modeled tumor responses were then all close to 11 log 10 of predicted tumor cell kill for a variety of different time dose schedules. This story will be told in this chapter. Although the numerical results of modeling depend to some extent on the values assumed for the parameters, ratios of parameters such as ratios and time dose trade-offs (grays per day) are often known sufficiently well for reasonable variations to lead to no clinically significant differences in predicted total dose or BED or NTD. In the modeling

5 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 7 described below, we take care to limit the assumed values of parameters to a small library of values selected from experience, avoiding elegant variation. Then, results that are useful and self-consistent are obtained. Unlike some other attempts at modeling, the number of initially viable cells per mm 3 is not essential in LQ modeling for time dose evaluations, because it is largely cancelled out against radiosensitivity in the standard BED formulation. This is also true of equivalent uniform dose (EUD; Niemierko 1997). Both BED and EUD enjoy similar stability for this reason. The most essential biological factor in LQ formulation is the ratio of the tissue concerned, which appears in the BED with weighting equal to dose per fraction. The other significant factors are the kick-off time of rapid tumor repopulation Tk and the doubling time of repopulating tumor cells Tp, which together with the value all appear in the repopulation term which is usually, but not always, a small proportion of BED. A glossary is attached at the beginning of this chapter of terms that are best for readers to know when reading about these topics. 1.2 The Simplest Modeling Years ago, mathematical models were regarded with suspicion, or with derision as playthings for children but not any more. Modeling has become an important scientific tool in the design and evaluation of topics from global warming to engineering design of aircraft and the pharmacological development of drugs, replacing expensive experimentation in many cases (Prof. Gordon Steel 1990, personal communication). With the aid of computers, mathematical modeling using optimization and new imaging are continuing to revolutionize treatment planning in radiotherapy, as other chapters in this book will show. There are good reasons for believing that the primary effects of radiation on biological tissues are cell damage and cell depopulation in renewing populations (Thames and Hendry 1987). This is still true whether the concern is damage to normal tissues or the elimination of every malignant cell in tumors. However, certain indirect biological end points, such as radiation sickness or extent of late fibrosis, do not appear to depend only on numbers of cells sterilized, although it does not mean that they are not a strong function of cells sterilized. The phantom of immunological response keeps rearing its head, with little practical effect. The additive effect of chemotherapy seems to apply most effectively when it is used concomitantly, and up to now amounts to some 10% of the total cell kill compared with radiotherapy. The strength of radiation as a treatment strategy is that it can and does reach wherever the physical plan puts it, with increasing efficiency of positional accuracy. The object of the present modeling is to find similar biological effects to the radiotherapy treatments with which we are familiar, from different scheduling of time and dose. That is the isoeffect modeling that has much history, passing through the cube root law of the 1930s, the Strandqvist slope of the 1940s, and the Nominal Standard Dose (NSD) or Time-dose Factor (TDF) of Dr Frank Ellis in the 1960s and 1970s and then to evaluate methods of doing better. We shall start by writing down some ideas for explaining the non-linear action of ionizing radiation in damaging biological cells. Radiation sterilizes cells, meaning that they do not die immediately, but at the time of the next cell division, or a few divisions later. An important factor is the repair occurring in the cells between irradiation and their next cell division. This repair in cells, i.e., recovery in tissues, depends on the turnover rate of renewing tissues a day or so for rapidly proliferating tissues and most tumors, but many months for the organs that normally proliferate slowly. That is the important difference in the tissues that LQ modeling helped to bring out, with the help of the famous or infamous ratio over The Seven Steps to LQ Heaven Brief Summary First, before explaining them in detail, we list here the Seven (algebraic) Steps to LQ Heaven, so that readers who wish to do so can skip several pages and turn to Section 1.4. That s where the story gets exciting. Alpha is the intrinsic radiosensitivity of the cells, defined as how many logs (to the exponential base e ) are killed (sterilized) per gray, in a nonrepairable way. Beta is the repairable portion of the radiation damage, requiring 6 h or more for complete repair. It can be regarded as the result of two charged-particle tracks passing through a sensitive target in the cell nucleus in less than 6 h, so this term has to be multiplied by d squared. E is the log e sum of

6 8 J. F. Fowler the non-repairable term and the partly repairable term. So for n fractions of d Gy dose each: E = n( αd+ βd 2 ) E = nd( α+ βd) E/ α = nd( 1+ dβ/ α) [1.1] [1.2] [1.3] Dividing through by to express as a ratio. E [1.4] α = nd + d ( 1 ) E = nd( 1+ d) α/ β With no repopulation considered; as for most types of late complication. This is the limited definition of BED. It applies to late effects. BED = E / = total dose RE where RE = (1 + d / [ ), and it is very useful. Next, we subtract the log cell kill due to repopulation of any cells during radiotherapy, after the kick-off or onset time Tk, where T is overall time and Tp is the average cell-number doubling time (in days) between Tk and T. E = nd ( + d) (T Tk) rate of repopulation per day [1.5] E = nd ( + d) log e 2(T Tk) / Tp [1.6] (log e 2 = 0.693) To transform the total log cell kill E into the total BED requires the same division throughout by that we carried out in step 3 (Eq. 1.3) above: BED = Ε d nd T Tk [1.7] α = 0, 693 ( 1+ ) ( α ) α / β Tp Final step = LQ heaven! BED can be expressed as Gy 3 (or Gy 2 ) for late complications, or as Gy 10 (or Gy x ) for tumor or early normal-tissue reactions, the subscript referring to the value used in its calculation. Gy 3 and Gy 10 values must not be mixed, as USA and Canadian or Hong Kong dollars cannot. However, several segments of a schedule can have their Gy 10 values added together and, separately, their Gy 3 values added, for a comparison of total BEDs amounting to a therapeutic ratio of Gy 10 Gy 3 representing tumor cell damage divided by late normal-tissue damage. This notation should always be used, or confusion quickly sets in. It both reminds us that this is a BED, not the real physical dose; and confirms which ratio was used, thus for which tissue each BED or EQD 2 Gy or NTD is calculated. We should state each time late or tumor or early BED or NTD (Fowler 1989). A further measure of radiation damage from these formulae is: Log e cell kill = E = BED, so that in the common log to base 10 scale, log 10 cell kill = log e cell kill/ To convert from BED to EQD 2 Gy or NTD (total equivalent dose in 2 Gy fractions): For late complications, divide Gy 3 by For tumor or early effects, divide Gy 10 by 1.2. The explanation comes from the identity BED1=BED2, where 1 is for d Gy per fraction and 2 is for 2 Gy per fraction, so for identical BEDs, we have: Total dose 1 RE1 = total dose 2 (RE for 2 Gy and the same ). NTD (1 + 2 / ) = BED. Then solve for NTD! 1.3 The Seven Steps to LQ Heaven The Details This is the section that experienced modelers might wish to skip by several pages, and go to Sections 1.4 and 1.5, where comparisons of actual schedules are tabulated and discussed. The mathematics of the LQ formula are very simple and are taught in courses before the end of the high school syllabus at age 15 or 16 years. Solving an algebraic quadratic equation and manipulating the conversion of an exponential to a logarithmic form are the only steps requiring an effort of memory and are only necessary if you get into more calculations than you will normally need. If you don t deal with this sort of arithmetic every week, you normally need no more than the four simplest keys (+,,, ) on your hand calculator, or even just the back of an envelope. The Seven Steps to LQ Heaven are intended to ensure that you, the reader, are never puzzled by such simple calculations again. Having followed, and understood, the seven steps that follow, you should be able to use them with confidence for any comparisons of radiotherapy schedules you wish to make in terms of BED (Fowler 1989, 1992). BED is proportional to log cell kill for cells of the specified ratio, and so is a strong function of biological effect. BED is itself a ratio (E/ ), the two

7 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 9 parameters of which are not individually important. Mathematically, E / is simply a linkage term showing equivalence between two schedules their isoeffect in the time-dose scale as well as providing a useful number, the BED. Its numerical value is a convenient number, representing a dose without the repairable component, that is avoiding dose per fraction or dose rate, until relative effectiveness (RE) is built in. We could talk about = 0.35 log e per Gy and E = 10 log 10 of cell kill, or equally = log e per Gy and E = one log 10 of cell kill, and the BED would be the same because it is E Development of the Simple LQ Formula E = nd (1 + d / / ) Cell depopulation is the main effect of radiation, both for eliminating tumors and for damaging normal tissues. In addition, some genes are activated, which can be relevant in those cells that survive irradiation, and some apoptosis (cell death independent of mitosis) may be caused. Although apoptosis does not appear to be the major effect of radiation, when it happens it adds to the effect. The promising strategy of damaging tumors by depleting their blood supply, with pharmacological or enzyme-pathway aid (Fuks et al. 1994), can be regarded as considering the neovasculature (that is, rapidly proliferating endothelial cells (Hobson and Denekamp 1984) as legitimate oncogenic targets, as well as the directly malignant tumor clonogens (Hahnfeldt et al. 1999). Radiation can reach them all. If a dose of radiation D sterilizes a proportion of cells in a given tumor or normal organ so that the number of viable cells is reduced from the initial No to Ns: the surviving proportion is Ns/No, which is designated S. This process is represented as S = e D/Do = exp ( D / Do), in its simplest form, where Do is the average dose that would sterilize one cell. This shows that the surviving proportion of cells is reduced exponentially with radiation dose. That is, each successive equal increment of dose reduces the surviving cells by the same proportion, not by the same number. The proportion of surviving cells would then decrease from 1 to 0.5, to 0.25, to 0.125, etc.; if higher doses per fraction were used, from 0.1 to 0.01 to Another way of writing this is: log e S = D / Do, which plots out as a graph of log e S vertically versus dose horizontally to give a straight line of slope minus D / Do. This would be called a single-hit curve. The dose Do would reduce survival by one log e which is to 37% survival. You can see that Do = 1 / in the LQ formula, that is 1 / (the dose to reduce cell survival by one log e ). Plotting graphically the proportion of surviving mammalian cells against single dose of radiation gives a curve, not a straight line. This curve starts from zero dose with a definite non-zero slope, gradually bending downward into a shoulder at less than 1 gray of dose, and continuing to bend downward until at least 10 or 20 grays. At higher doses, the cell survival curve appears to become nearly straight again but of course steeper than at the origin (Gilbert et al. 1980). This is the well-known cell survival curve where the logarithm of the surviving proportion of cells is plotted downward, against radiation dose on the horizontal axis (Fig 1.1). It is the same curve as the negative logarithm of the fraction of cells killed actually sterilized so that they die later, after the next cell division or a few divisions. This is the plot of log e proportion of cells surviving: S = d d 2. Therefore also: log e proportion of cells sterilized (killed) = + d + d 2. It is this logarithm of the number of cells sterilized that can be divided into one part proportional to dose d; and another part proportional to dose- Fig The simple cell survival curve for linear quadratic cell kill versus radiation dose, for a single dose of radiation delivered within a few minutes. The alpha component increases as shown linearly with dose. The beta component is added to this in a curving pattern, increasing with the square of the dose. This example is numerically correct for the ratio of 3 Gy

8 10 J. F. Fowler squared d 2 (that is quadratically, where two sublesions combine, each produced in number proportional to dose). The logarithm (proportion) of lethal events caused by a dose d is then: E = d + d 2 The linear component is found to be not repairable beyond a few milliseconds after the irradiation, but this does not mean it cannot be altered, by oxygen if present at the time of irradiation as one major example. However, the dose-squared damage gradually fades over a few hours. It is repaired by several processes within the biological cells, mostly within the DNA, so that cell survival recovers toward the straight initial slope. Cells and tissues are said to recover and biochemical lesions in DNA are said to be repaired. Let us call this cell-number damage E, the logarithm of the number of cells sterilized by a dose d in grays (Gy). Then, we just write the first equation as: E = d + d 2 where and are the coefficients of the linear component and the dose-squared component, respectively. This is the first step on the ladder to explaining the LQ formulation, commonly taught as the Seven Steps to Heaven. If you remember this little starting formula you will have no trouble at all with the next three steps. LQ Heaven is reached when you understand the LQ steps well enough not to forget them within the next few days. The next step is the simple one of adding several fractions of daily doses, each of d grays, to obtain the total dose if n fractions (daily doses) are given. Figure 1.2 illustrates the sequence of equal fractions, giving a total curve that is made up of a sequence of small shoulders, in toto an exactly linear locus, of slope depending on the value of Eq. 1.1 at each dose d: E = n( αd+ βd 2 ) [1.1] The dose per fraction d can be taken outside the parentheses, nd being of course the total dose: E = nd( α+ βd) [1.2] We usually know the ratio of the two coefficients,, for given cells and tissues much better than we know their individual values, so the next step is to express E in terms of this ratio. It is most usefully done by dividing both sides of Eq. 1.3 by (Barendsen 1982). If instead we divided by, then the resulting BED would be in terms of dose squared, which would be awkward. E / = nd (1 + d ) [1.3] which is also of course identical to: E [1.4] α = nd + d ( 1 ) E = nd( 1+ d) α/ β Because is defined as a number (log number of cells sterilized) per gray, the term E / has the dimensions of dose. It is in fact the BED that we are seeking to calculate, so Eq. 1.4 gives it to us. We are halfway up the Steps to Heaven and this equation enables us to do many useful things in predicting biological damage (Fowler 1989; also Chapters 12 & 13 in Steel 2002) Biologically Effective Dose Fig The survival curve for four equal radiation doses given sequentially, with sufficient time at least 6 h between them to allow complete repair of the beta component of radiation damage. Since the shape of each is then a repetition of the previous dose, the track of the result after each dose fraction is a straight line when plotted as log cell kill against dose as shown The basic concept of BED was defined by Barendsen (1982), who at first called it extrapolated tolerance dose (ETD), meaning that dose which, if given as an infinite number of infinitely small fractions (along the initial slope of the cell survival curve), or at a very low dose rate (so that all the quadratic damage has been repaired), would cause the same log cell kill

9 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 11 as the schedule under consideration, thinking of the maximum dose that a normal tissue would tolerate. Since it was obvious that this conceptual extrapolation to very small dose per fraction could be applied to any level of damage, not just to the maximum tolerated level or only to normal tissues, it was soon renamed extrapolated response dose (ERD) and later to the more general BED (Fowler 1989). It is illustrated graphically in Figure 1.3. Because BED is defined in relation to the initial slope, i.e., the linear component of damage, it is E/ Because has the dimensions of 1/dose, E/ has the dimensions of dose, as we require. We are talking here about an averaged value of during the weeks of radiotherapy. Different values of could be applied to different segments of a schedule but there is not yet evidence to justify this. Since the definition of BED is the ratio E/, the individual values of E and are irrelevant for estimating relative total doses. The ratio E/ is mathematically a link function, signifying biological equivalence between two schedules having equal effect. The individual values of E or have no necessary biological significance in LQ formulation. Values of are particularly vulnerable to variation of tumor size, stage of tumor, and accuracy of dose, but provided that the ratio between E and (and does not vary between one prospective population in a clinical trial and another there are no effects on ratios of doses between schedules, which are what we want to compare. This is the important reason why BED, like EUD (Niemierko 1997), is robust. They are relatively stable if parameters are varied by modest amounts. These biological ratios between log cell kill and cellular radiosensitivity have in fact been determined in certain biological experiments concerning skin and intestinal clones in mice (Withers 1967, 1971), so these tissues can be put on to a more exact basis concerning number of cells per millimeter of mucosal surface. However, the use of a number to designate log cell kill in tumors depends on a specific assumption of how many viable malignant cells were present per mm 3 of tumor volume, which is usually unknown. The number has wildly varied between ten thousand million (10 10 ) and one-tenth per tumor among various authors for various tumors (the latter being curable with 90% probability). Because of the real variability between tumors and patients, a value for [derived for example from a gamma- 50 slope of dose versus tumor control probability (TCP)] cannot be regarded as a reliable guide to cell numbers, as some writers have assumed and argued about. Heterogeneity decreases the extracted value of from populations. What is interesting is that, as more detailed and precise descriptions for stage of tumors are developed, as they are by various international agreements, the recorded gamma-50 slopes from clinical data become steeper (or values higher), illustrating less heterogeneity (Hanks et al. 2000; Regnan et al. 2004). The log 10 cell kill values quoted in the tables below are all assuming =0.35 log e per Gy Relative Effectiveness Fig Graphical illustration of the concept of biologically effective dose. The dashed line represents the log cell kill of 20 (or more) equal fractions of radiation. BED is the dose that would cause the same log cell kill E as the schedule with n d fractions, if it could be delivered in an infinite number of infinitely small fractions, or at very low dose rate. All the beta component would then have been repaired, and the straight line representing the log cell kill versus total dose would follow just the initial slope alpha as shown, its slope determined by the value of alpha only. The ratio of BED to the real physical dose is of course the RE, relative effectiveness=bed total dose The BED thus comprises the total dose nd multiplied by a parenthesis (1 + d ). This parenthetic term is called the RE, and is one of the most useful concepts of the LQ formulation. RE is what determines the strength of any schedule when multiplied by total physical dose. BED is simply total dose multiplied by RE: BED = nd (1 + d ) = nd RE [1.4] or BED = total dose RE

10 12 J. F. Fowler When dose rate varies, the term containing includes, instead of just d, the dose rate and the recovery rate of tissues as a ratio, as discussed in detail elsewhere (Barendsen 1982; Thames and Hendry 1987; Fowler 1989). The concern of RE is obviously dose-per-fraction size, in relation to the ratio for different tissues. RE is larger for larger dose per fraction and for smaller ratios of. Dose per fraction is therefore one of the three major factors in LQ formulation, the co-equal second being the ratio. The third is overall time, which we deal with in the remaining three steps to LQ heaven below. The major biological importance of RE depends on the fact that is large for rapidly proliferating tissues such as most tumors and small for slowly proliferating tissues such as late complications. This is attributed to the fact that slowly proliferating tissues, with long cell cycle times, have more time to carry out repair of radiation damage than do short cell cycles. The exact molecular causes of these differences in ratio of radiosensitivity to repair are not yet known but inhibition of epidermal growth factor receptors (EGRFs) is likely to be involved Fowler 2001). It is these differences that enable conventional radiotherapy to succeed, often using a large number of small fractions such as 30 or 35F 2 Gy. We shall discuss the differences in in the next section. RE is of course high when d is high and is low, for example as in hypofractionation (fewer and larger fraction sizes than normal), which could therefore cause excessive damage in slowly responding, late-reacting tissues, i.e., in late complications. In contrast, RE is naturally low for hyperfractionation (more and smaller fractions) and for low doserates (lower than 100 cgy/h; ICRU can be criticized for defining low dose rates as up to 200 cgy/h, because their BEDs differ significantly from those for 50 or 100 cgy/h). It is the difference in doses per fraction above or below 2 Gy that has biased radiotherapy in the direction of many small fractions. Total doses at 2 Gy per fraction should be above 60 Gy for cure of all except a few radiosensitive types of tumor; and preferably above 70 to 90 Gy NTD 2 Gy for many tumors of stage II or III sizes. Equations 1.4 and 1.4a enable a great many aspects of radiotherapy to be compared simply and quickly. A question often asked is what new total dose should be used if dose per fraction is changed from d 1 to d 2? This is easily answered by taking the inverse ratio of the two values of RE. If the two total doses are D 1 and D 2, of which D 1 is known and D 2 is the value sought, both of the BEDs have to be equal, by definition: BED=total dose RE = D 1 RE 1 = D 2 RE 2 ; D2 RE2 (1 + d1/α/β) So that = = D RE (1 + d /α/β) 1 [1.4b] This is an example of a comparison of schedules where the only biological factor is the ratio, treated as if it were a single parameter Overall Treatment Time 1 The major effects of radiation are diminished by any proliferation occurring in the cell populations during the weeks of radiotherapy, obviously by replacement of some of the cells that were sterilized by irradiation. In slowly proliferating tissues, naturally those in whom the reactions appear late, this replacement by repopulation is negligible (as in nerve tissue) (Stewart 1986). In rapidly proliferating tissues, however, including most tumors, it can counteract up to one-third of the cell-killing effect of the radiotherapy, although there is a delay of some days before it begins in tumors; it is more rapid in normal mucosa, at 7 days in oral mucosa. This delay in tumor repopulation is because the tumor needs time to shrink due to the treatment, so as to bring more of the surviving cells into range of the generally poor blood supplies in the tumor. In solid tumors above a few millimeters in diameter, the daily production of cells is much faster than the volume growth rate would suggest. In most carcinomas, there is a cell loss factor of 70 95% due to nutritional failure and, to a lesser extent, due to apoptosis. This causes the volume doubling times of tumors to be as slow as months, although the number of cells born would cause the population to double in only a few days if no such cell loss occurred. Therefore, the clinically observed doubling time of a tumor is no guide to the cell repopulation rate inside the tumor. This is one of the curious facts in cancer that has only been apparent from kinetic population studies in the 1970s, using originally radioisotope labels and recently immunological fluorescent labels with flow cytometry (Begg and Steel 2002). Most human carcinomas have volume doubling times varying from 1 month to 3 months, but their cell population birth rates yield cell-number doubling times of 2 10 days, with 3 5 days commonly found. The cell doubling times are called Tpot (potential doubling times, meaning cell birth rate in the absence of nutritional or apoptotic cell loss ). Prostate tumors are exceptionally slowly proliferating with median cell doubling times of 42 days (range 15 days to >70 days). 2

11 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 13 So, if a number S of cells survives a certain phase of treatment, this number will increase at the rate of Se +1/Tp per day and Se +t/tp in t days. This means the number of cells increases by a constant proportion per day. This rate is conventionally described as the time in days required to double the number of cells, Tp, so the population increases by log e 2/Tp per day. Therefore, the log e number of cells killed decreases by log e 2 Tp per day, meaning its change is 0.693/Tp per day and t/tp in t days. It is very important that we do not forget the delay in start of repopulation mentioned above, after a kick-off or onset time designated Tk days. Allowing for this, the time available for repopulation is of course t = T Tk days, where T is the overall time in days. It is important to note that the treatment begins at day 0, not day 1. The log e amount of repopulation is then 0.693(T Tk) / Tp, and it has to be subtracted from the log e cell kill E that we calculated above in Eq Thus, the total amount of log cell kill, allowing for repopulation, is of course: E = nd ( + d) (T Tk) rate of repopulation per day [1.5] E = nd ( + d) log e 2(T Tk) / Tp [1.6] where T = overall treatment time (in days, the first day being 0 not 1), Tp = cell population doubling time during treatment, Tk = starting time of repopulation, and Tp = average cell-number birth rate during the irradiation. Note that Tp may be different from Tpot, which can only be measured before any treatment is given; Tp is possibly somewhat faster. This simple modeling, assuming zero repopulation up to Tk days and a constant doubling time Tp after Tk, is certainly an approximation to a smooth curve of accelerating Tp, but several more complex models to add different times at which faster Tp is assumed have not succeeded in giving better correlations with clinical outcome (van Dyk et al. 1989; Denham and Kron 2001), although Denham and Kron (2001) made a persuasive case for some type of accelerating curve in tumors. Fenwick has recently proposed an ingenious Delayed Feedback (from the level of cell depletion reached) algorithm which deals elegantly with this concept (Fenwick 2006). To transform E into the total BED requires the same division throughout by that we carried out in step three (Eq. 1.3), and this is the final equation, the seventh step: Putting it into words: BED = E = Total Dose Relative Effectiveness α minus (loge 2) (Days availible for repop) alpha Cell Doubling Time [1.7a] It will be noticed that this LQ formula now involves three more parameters, so we might think that LQ has lost its innocence when repopulation is included. A value must be chosen for radiosensitivity and the start of rapid repopulation in tumors Tk, and for the doubling time Tp of viable cells thereafter. There are, however, some clinical data that enable ratios of these parameters to be estimated. (There is a practical snag in that it is easy to forget the in the last term! But, if you remember that this repopulation term is usually about Gy 10 per day for rapidly proliferating tissues including most tumors, you can easily make a rough check on the subtraction from the simpler BED of step four (Eq. 1.4). This term is somewhat greater if doses per fraction are larger than 2 Gy.) In radiotherapy practice, late-responding normal tissues generally have a normal turnover time of many months, and these are reflected first of all in their kick-off time of acceleration, Tk, after starting radiotherapy (Stewart and van der Kogel 2002), and also obviously in the value of Tp after compensatory repopulation has started (Hendry et al. 1996; Fowler and Chappell 2000), which may be a little shorter than Tpot measured, as it can only be, before any treatment. It is this repopulation aspect of radiobiological modeling that has brought most radiotherapy departments to pro-actively avoid gaps in treatment, and in many to choose shorter schedules (Hendry et al. 1996). To a radiobiology modeler, it is surprising that this took so long to realize (Fowler 1978; Withers et al. 1988; Fowler and Lindstrom 1982; Hendry et al. 1996). The huge difference between gross tumor volume doubling time and the cell birth doubling time, Tpot, is due to the high cell loss factor (70 90%) in carcinomas. The kick-off time for tumor repopulation, Tk, could be between 21 days and 32 days in human head and neck tumors (Brenner 1993; Roberts and Hendry 1999) Acute Mucosal Tolerance BED E nd (1+ d 0, 693 = = ) ( T Tk) α α / β αtp [1.7] A particular application of Eq. 1.7 above has been a long-term development of a collection of data about

12 14 J. F. Fowler head and neck radiotherapy schedules that are judged to be clinically just tolerable from the acute reaction point of view. A comprehensive review of published head and neck schedules with consequential recommendations for a tolerance zone of early BED was recently published by the present author and colleagues (Fowler et al. 2003c). The best matching with these not particularly small treatment fields, for nearly intolerable acute mucosal reactions in many types of fractionation regime, was found to be at BED = Gy 10 if the following specific parameters in Eq. 1.7 were used: = 10 Gy; = 0.35 Gy per log e ; Tk (starting time of repopulation) = 7 days for oral mucosa (Dörr et al. 2002); and Tp = 2.5 days (average mucosal cell doubling time during irradiation). There is provisional information that the range of BED may be somewhat higher in rectal mucosa, for example Gy 10 together with a mucosal area limitation down to a few centimeters squared for doses of Gy NTD (Huang et al. 2002; Vargas et al. 2005), but further confirmation is required. A practical conclusion from our acute mucosal review (Fowler et al. 2003c) was that if a new schedule causes too strong acute reactions, these can be avoided easily by extending the overall time by a few days. Acute BED for mucosa is normally reduced at a rate of about 0.8 Gy 10 per day (with 2 Gy fractions), after the time Tk = 7 days from starting daily radiotherapy (Dörr et al. 2002) To Convert from BED to NTD or EQD 2 Gy We have talked in terms of BED in Gy 3 and Gy 10 mainly, and familiarity with these terms and numbers is recommended. However, those who do not use them regularly are often more comfortable with their conversion into the biological equivalent of total dose in 2 Gy fractions. This conversion of BEDs into EQD 2 Gy or NTD (normalized total dose/normalized to 2-Gy fractions; Maciejewski et al. 1986; Joiner et al. 2002) is a further mathematical step, which experienced modelers may not wish to do, but it is very simple, as described below. It does, and this is important, give different EQD and NTD values for late complications from the EQD and NTD for tumors in the same schedule, so that subscripts for BEDs in Gy 3 or Gy 10, etc. must be kept rigorously through every step of every calculation. Further, the translation from tumor BED or late complications BED into tumor NTD or late complications NTD must be made when the results are summarized. Such designation of Gy 3 or Gy 10, and Late or Tumor or Acute is necessary every time, or readers (and you the writer!) will quickly become confused. We can see, from the obvious equality of: BED of any schedule 1, called BED1, required to be equal in radiobiological effect to BED2, shows that: total dose D1 the RE for its dose per fraction must be equal to NTD 2 Gy RE for 2 Gy fractions, with the appropriate ratio so we obtain the formula: NTD 2 Gy RE2 = BED1; therefore NTD 2 Gy = BED1 / RE2 = BED1 (RE for 2 Gy fractions). QED Therefore, an NTD 2 Gy or EQD 2 Gy can be derived from any BED by simply dividing that BED by the RE for 2 Gy fraction size and the same ratio as used to calculate the BED (Table 1.1) One Example First let us look at the simple example of a 60 Gy dose given with different fraction sizes, starting with 2 Gy 30F in 6 weeks. The late complications BED is usually calculated first, assuming = 3 Gy, and from Eq. 1.4 it is: BED = nd (1 + d / ( )) 60 (1 + 2 / 3) = = 100 Gy3 The tumor response BED is calculated assuming = 10 Gy and is therefore: 60 (1 + 2 / 10) = = 72 Gy10 Both the late BED of 100 Gy 3 and the tumor BED of 72 Gy 10 correspond to an NTD to 2 Gy fractions or EQD 2 Gy of 60 Gy, of course. Table 1.1. Conversion of BED to NTD or EQD 2 Gy Type of tissue for BED Divide the BED by: Late complications 3 Gy (1 + 2 / 3) = Early and tumor effects 10 Gy (1 + 2 / 10) = 1.2 Exceptions would be: Late CNS with = 2 Gy 2 Gy (1 + 2 / 2) = 2.0 Prostate tumors = 1.5 Gy 1.5 Gy (1 + 2 / 1.5) = 2.333

13 Practical Time-Dose Evaluations, or How to Stop Worrying and Learn to Love Linear Quadratics 15 For a stronger schedule of 70 Gy = 2 Gy 35F the corresponding late (complications) BED and the early (tumor) BED would be = 117 Gy3 and = 84 Gy10, respectively. Both the late BED of 117 Gy3 and the tumor BED of 84 Gy10 correspond of course to the NTD or EQD 2 Gy of 70 Gy. Those four values of BED form useful reference points when considering other schedules. In particular, the late complications BED of 117 Gy3 should not be exceeded except in special circumstances, such as research or IMRT-related new schedules. The NTD can be obtained by dividing the BED by for late complications or by 1.2 for the early or tumor effects, respectively, as described in Table 1.1 above. Other examples have been worked out in detail in the Appendix (Section 1.8), starting with 60 Gy in 4 Gy fractions What is the Standard of Precision of These Estimates of BED or NTD? Gamma Slopes How much margin of tolerance can we allow before being concerned about an overdose to normal tissues, above the arbitrary but currently practical limit of 117 Gy 3? And how much should we be concerned about tumor BEDs below the 70 Gy NTD or EQD 2 Gy, that is below a BED of 84 Gy 10? These questions require whole papers to answer them properly, involving the slopes of dose response curves. These slopes are often called gamma-50 or gamma-37, a term derived from photographic film density, meaning a change in absolute percentage response for a 1% change in total dose or NTD (and BED if fraction size is not changed). However, a common sense answer is that about a 7% change in BED or total dose at 2 Gy fraction size would probably be large enough to be clinically noticeable, and a 3% change would probably not, until very many hundreds of patients had been treated in the same way. The degree of anxiety about an over- or under-dose would also depend on the actual circumstance of what risk is most important for the individual patient. Gamma-50 or gamma-37 is used to define the steepest part of a dose response curve, which is at 50% probability for logistic modeling, but at 37% for Poisson probability modeling. The difference is rarely important in comparison with the precision of clinical results. For many types of tumor gamma- 50 or -37 slopes found are about 1.5 to 2 times the percentage change in total dose (and BED if dose per fraction is unchanged). This is in the middle range of the sigmoid response curve between about 20% and 70% for local control (LC). So tumor response often corresponds to a gamma slope of 1.5 or 2. For doses above or below that middle range, the slopes will of course be smaller. For certain types of tumor that have been well classified in detail so that narrow risk categories are available, the change of response with BED may be greater than twice the percentage (Hanks et al. 2000; Regnan et al. 2004). In general a 10% change in tumor or NTD 2 Gy or EQD means a change in LC of approximately 20%. Wadsley and Bentzen (2004) showed that there was a highly statistical relationship (P= ) between locoregional control (LRC) and later overall survival in head-and-neck cancer, 5-year overall survival averaging two-thirds of the 2-year LRC. For normal tissues, the dose response curves are often steeper than for tumors, because normal tissues are less heterogeneous in physiological properties than tumors are. Gamma-50 slopes of three or four in the steepest range of late complications are found, although slopes are shallower at the lower incidences of a few percent that are common for serious complications; the toe of the curve (and at the top, should we see any result close to 90% or 100%). 1.4 Rejoining Point for Those Who Skipped: How to Evaluate a New Schedule Brief Summary In case some of these details might have slipped your mind, we now summarize the three main weapons in our armamentarium for assessing any proposed schedule, before irradiating any patient. For example, we may wish to add an introductory phase-i dose-escalation study of effects in normal tissues before deciding on a particular dose level in a long clinical trial. A calculation of BEDs or NTDs would enable any arm to be evaluated theoretically in relation to known schedules. Such theoretical comparisons do not replace phase-i clinical trials, but they can put them into critical perspective and save time-wasting steps with dose intervals that may be too wide or too narrow. Calculate late complications BED, = 3 Gy, with no time factor [Eq.1. 4], keeping below 117 Gy 3. Estimate tumor BED including repopulation, as Gy10 [Eq. 1.7]. For head-and-neck or lung tumors assume: